Electrons & Phonons Ohm’s & Fourier’s Laws Mobility & Thermal Conductivity Heat Capacity Wiedemann-Franz Relationship Size Effects and Breakdown of Classical Laws

Thermal Resistance, Electrical Resistance P = I2 × R ∆T = P × RTH ∆ V = I × R R = f(∆T) Fourier’s Law (1822) Ohm’s Law (1827)

Poisson and Fourier’s Equations drift diffusion diffusion only Note: check units! Some other differences: Charge can be fixed or mobile; Fermions vs. bosons…

Mosquitoes on a Windy Day Some are slow Some are fast Some go against the wind n n(v0)dv wind v dv Also characterized by some spatial distribution and average n(x,y,z) Can be characterized by some velocity histogram (distribution and average)

Calculating Mosquito Current Density Area A dr Current = # Mosquitoes through A per second NA = n(r,v)*Vol = n(r,v)*A*dr Per area per second: JA = dNA/dt = n(dr/dt) = nv

Charge and Energy Current Density Area A dr What if mosquitoes carry charge (q) or energy (E) each? Charge current: Jq = qnv  Units? Energy current: JE = Env  Units? units are C/m2/s = A/m2 and J/m2/s = W/m2

Particles or Waves? Recall, particles are also “matter waves” (de Broglie) Momentum can be written in either picture So can energy Acknowledging this, we usually write n(k) or n(E)

Charge and Energy Flux (Current) Total number of particles in the distribution: Charge & energy current density (flux): Of course, these are usually integrals (Slide 11) label density of states, and occupation probability function

What is the Density of States g(k)? Number of parking spaces in a parking lot g(k) = number of quantum states in device per unit volume How “big” is one state and how many particles in it? L “d” dimensions L “s” spins or polarizations L

Constant Energy Surface in 1-, 2-, 3-D ky kz k kx kx ky kx ml ≈ 0.91m0 mt ≈ 0.19m0 Si For isotropic m*, the constant energy surface is a sphere in 3-D k- space, circle in 2-D k-space, etc. Ellipsoids for Si conduction band Odd shapes for most metals

Counting States in 1-, 2-, or 3-D ky kz k kx kx ky kx System has N particles (n=N/V), total energy U (u=U/V) This is obtained by counting states, e.g. in 3-D:

What is the Probability Distribution? Probability (0 ≤ f(k) ≤ 1) that a parking spot is occupied Probability must be properly normalized Just like the number of particles must add up But people generally prefer to work with energy distributions, so we convert everything to E instead of k, and work with integrals rather than sums

Fermi-Dirac vs. Bose-Einstein Statistics ∞ T=0 T=0 T=300 T=300 T=1000 T=1000 Fermions = half-integer spin (electrons, protons, neutrons) Bosons = integer spin (phonons, photons, 12C nuclei) In the limit of high energy, both reduce to the classical Boltzmann statistics:

Temperature and (Non-)Equilibrium Statistical distributions (FD or BE) establish a link between temperature, quantum state, and energy level Particles in thermal equilibrium obey FD or BE statistics Hence temperature is a measure of the internal energy of a system in thermal equilibrium What if (through an external process) I greatly boost occupation of electrons at E=0.1 eV? Result: “hot electrons” with effective temperature, e.g., at T=1000 K Non-equilibrium distribution T=300 K T=1000 K

Counting Free Electrons in a 3-D Metal # states in k-space spherical shell spin Convert integral over k to integral over E since we know energy distribution (for electrons, Fermi-Dirac)

Counting Free Electrons Over E At T = 0, all states up to EF are filled, and above are empty This also serves as a definition of EF E EF So EF >> kBT for most temperatures

Density of States in 1-, 2-, 3-D (with the nearly-free electron model and quadratic energy bands!) van Hove singularities

Electronic Properties of Real Metals n (1028 m-3) m*/m0 vF (106 m/s) Cu 8.45 1.38 1.57 Ag 5.85 1.00 1.39 Au 5.90 1.14 Al 18.06 1.48 2.02 Pb 13.20 1.97 1.82 Fermi equi-energy surface usually not a sphere source: C. Kittel (1996) In practice, EF and m* must be determined experimentally m* ≠ m0 due to electron-electron and electron-ion interactions (electrons are not entirely “free”) Fermi energy: EF = m*vF2/2

Energy Density and Heat Capacity similar to before, but don’t forget E(k) For nearly-free electron gas, heat capacity is linear in T and << 3/2nkB we’d get from equipartition Why so small? And what are the units for CV?

Heat Capacity of Non-Interacting Gas Simplest example, e.g. low-pressure monatomic gas Back to the mosquito cloud From classical equipartition each molecule has energy 1/2kBT per degree of motion (here, translational) Hence the heat capacity is simply Why is CV of electrons in metal so much lower if they are nearly free?

Current Density and Energy Flux What if f(k) is a symmetric distribution? Energy flux: Check units?

Conduction in Metals Balance equation for forces on electrons (q < 0) Balance equation for energy of electrons Current (only electrons near EF contribute!)

Boltzmann Transport Equation (BTE) The particle distribution (mosquitoes or electrons) evolves in a seven-dimensional phase space f(x,y,z,kx,ky,kz,t) The BTE is just a way of “bookkeeping” particles which: move in geometric space (dx = vdt) accelerate in momentum space (dv = adt) scatter Consider a small control volume drdk Rate of change of particles both in dr and dk Net spatial inflow due to velocity v(k) Net inflow due to acceleration by external forces Net inflow due to scattering = + +

Relaxation Time Approximation (RTA) Where recall v = and F = In the “relaxation time approximation” (RTA) the system is not driven too far from equilibrium Where f 0(E) is the equilibrium distribution (e.g. Fermi-Dirac) And f’(E) is the distribution departure from equilibrium RTA: assume the distribution function has two parts: a large component shaped like the equilibrium function, and a small perturbation obtained by solving the BTE

BTE in Steady-State (∂f/∂t = 0) Still in RTA, approximate f on left-side by f 0 Furthermore by chain rule: So we can obtain the non-equilibrium distribution And now we can directly calculate the current & flux

Current Density and Mobility This may be converted to an integral over energy Assume F is along z: (dimension d=1,2,3) (Lundstrom, 2000) (Ferry, 1997) Assumption: position-independent relaxation time (τ)

Energy Bands & Fermi Surface of Cu Fermi surface of Cu is just a slightly distorted sphere  nearly-free electron model is a good approximation

Why the Energy Banding in Solids? Key result of wave mechanics (Felix Bloch, 1928) Plane wave in a periodic potential Wave momentum only unique up to 2π/a Electron waves with allowed (k,E) can propagate (theoretically) unimpeded in perfectly periodic lattice

What Do Bloch Waves Look Like? Periodic potential has a very small effect on the plane-wave character of a free electron wavefunction Explains why the free electron model works well in most simple metals

Semiconductors Are Not Metals empty states E empty conduction band EF filled valence band electron states in isolated atom electron states in metal electron states in semiconductor Filled bands cannot conduct current What about at T > 0 K? Where is the EF of a semiconductor?

Semiconductor Energy Bands Si GaAs Equi-energy surfaces at bottom of conduction band Si GaAs Silicon: six equivalent ellipsoidal pockets (ml*, mt*) GaAs: spherical conduction band minimum (m*)